Summary Analysis

Coin catalogue section:	Kelenderis
Coin corpus datasets:	Kelenderis, Third staters

Summary

Third staters can be divided into two groups. The first consists of Types 4.1–2 (Group 4A) and the second of Types 4.3–10 (Group 4B), with the second group characterized by a slightly lower weight standard than the first group and at the same time a higher dispersion of coin weights. The difference in the average observed weights of these two groups is 0.09 g, and the difference in medians is 0.07 g. However, due to the small number of observations, this segmentation of third staters cannot be considered completely certain and the only possible one, and a larger sample of data is needed to confirm or refine it.

Analysis

Box plots1 of individual coin types and basic descriptive statistics are presented in Figure 1 and Table 1 (Std. Dev. denotes the standard deviation and IQR the interquartile range), respectively. Type 4.5 is represented by two specimens, one of which has an exceptionally low weight of 2.81 g (coin no. 17), which is quite different from the weights of the other coins. This coin, however, appears to be well preserved and there is no reason to doubt its authenticity and has therefore not been excluded from this weight analysis.

Figure 1: Box plots of individual coin types

Type	Count	Mean	Median	Std. Dev.	IQR
4.1	1	3.59	3.59	0.00	0.00
4.2	5	3.55	3.56	0.04	0.07
4.3	3	3.52	3.50	0.04	0.05
4.4	7	3.48	3.50	0.09	0.14
4.5	2	3.19	3.19	0.53	0.75
4.6	3	3.47	3.46	0.08	0.12
4.7	1	3.45	3.45
4.8	10	3.49	3.49	0.09	0.10
4.9	1	3.48	3.48
4.10	1	3.49	3.49

Table 1: Basic descriptive statistics of coin types

As Figure 1 and Table 1 show, the weight standard of Types 4.1–2 aappears to be higher than that of the other types. Third staters can therefore be divided into the following two groups:

Group 4A:	Types 4.1–2;
Group 4B:	Types 4.3–10.

However, due to the small size of the data sample, this segmentation is still preliminary. Note, for example, that weights of two coins of Type 4.8 exceed the highest weight observed in Group 4A. These are coins nos. 28 and 31 weighing 3.64 g and 3.62 g, respectively, while the heaviest coin of Group 4A weighs 3.61 g (coin no. 6). Perhaps this can be explained by the fact that less attention was paid to the weight of the flans of this type.

Table 2 shows the descriptive statistics of all third staters and of Groups 4A and 4B.

Statistics	All coins	Group 4A	Group 4B
Number of coins:	34	6	28
Mean:	3.48	3.56	3.47
Standard deviation:	0.14	0.04	0.15
Interquartile range:	0.10	0.08	0.10
Skewness:	-3.37	-0.17	-3.18
Kurtosis:	17.07	1.59	15.05
Minimum:	2.81	3.51	2.81
25th percentile:	3.46	3.51	3.44
Median:	3.50	3.56	3.49
75th percentile:	3.56	3.59	3.54
Maximum:	3.64	3.61	3.64

Table 2: Descriptive statistics of coin groups

The following charts visualize the weight distributions of these groups. Figure 2 shows box plots and Figure 3 present relative frequency histograms (the bars represent the relative frequencies of observations ranging from 2.80 to 3.65 g in increments of 0.05 g). The continuous curves represent approximations of the data by the Weibull distribution2 based on maximum likelihood estimates. Cumulative distributions are shown in Figure 4.

Figure 2: Box plots of Groups 4A and 4B

Figure 3: Relative frequency histograms of Groups 4A and 4B

Figure 4: Cumulative distributions of Groups 4A and 4B

The two-sample Kolmogorov-Smirnov test rejects the hypothesis of the equality of the weight distributions of groups 4A and 4B (p-value of 0.018). Similarly, the one-sided Welch’s t-test3 rejects the null hypothesis that the mean weights of both groups are equal in favour of the alternative that the mean weight of Group 4A is higher than that of Group 4B (p-value of 0.004). The percentile bootstrap method was also used to supplement these results. Table 3 shows the observed sample medians and bootstrap 95% confidence intervals.4 Table 4 shows the difference in sample medians, its bootstrap 95% confidence interval and p-value.5 Overall, we can conclude that the weight distributions of groups 4A and 4B are significantly different at the 5% significance level, but due to the small number of observations, this segmentation of third staters cannot be considered completely certain and the only possible one.

	median	95% confidence interval
Group 4A	3.56	3.51	3.60
Group 4B	3.49	3.46	3.51

Table 3: Medians and their confidence intervals

	medians difference	95% confidence interval		p-value
Group 4A vs Group 4B	0.07	0.01	0.12	0.009

Table 4: Differences in medians